Inverse Problems · 1988 · 26 citations · 7 references
Natural SciencesGeophysicsPhysicsMathematical PhysicsQuantum MechanicsTheoretical PhysicsFourier AnalysisIntegral Equation FormulationsIntegral EquationsIntegral GeometryScattering ProblemsScattering TheoryDiffraction TomographyFourier Inversion FormulaInverse ScatteringMigration AlgorithmsQuantum ScatteringRadon TransformSeismic MigrationTomography
Abstract. Formulae involving double integrals over spheres arise naturally in inverse scattering problems since the scattered data are measured in the space R x Sz x Sz. In this paper we derive a relation between differential forms on the space S" ' x S"- ' , and those on the space Z x S n-2 x S where Z is a real interval. Specifically, d ( dq = sin" ' 6 d6 d $ dv (t, 7) E S '- ' x S '- ' and (6, $, U) E Z x S"-2 x S"-'. This allows us to derive the results of John relating the iterated spherical mean of a function to its spherical mean in a simple way; to obtain new inversion formulae for the Fourier and Radon transforms; to extend formulae for linearised inverse quantum scattering and diffraction tomography to the multifrequency case; and also to establish a relation between multifrequency diffraction tomography and seismic migration algorithms. 1.
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